closed door machining-error compensation of complex surfaces

Original manuscript submitted. Final version of the article now published in the International Journal of Computer Integrated Manufacturing, Taylor and Francis, DOI:10.1080/0951192X.2013.874577. Published online: 10 Feb 2014, available for download at http://www.tandfonline.com/eprint/zZqcVcR6zS3QeZVxIjPN/full

Closed door machining-error compensation of complex surfaces using

the cutting compliance coefficient and on-machine measurement for a

milling process

R.Guiassaa, J.R.R. Mayera*, M. Balazinskia, S. Enginb and F.-E. Delormeb

a)Département de génie mécanique, École Polytechnique de Montréal, P.O. Box 6079, Station

Downtown, H3C 3A7 Montréal (QC),b)

Pratt and Whitney Canada Corp., 1000 Marie-Victorin

Longueuil, Quebec, Canada J4G 1A1

* Correspondingauthor: René Mayer,École Polytechnique de MontréalC.P. 6079, succ.

Centre-ville, Montréal (Qc) CANADA H3C 3A7

Tel. 514-340-4711, ext : 4407

Fax 514-340-5867

e-mail: [email protected]


Closed door machining-error compensation of complex surfaces using

the cutting compliance coefficient and on-machine measurement for a

milling process

Abstract

Closed door machining is a strategy for producing a part within tolerance using

on-machine measurement and automatic process adjustment as opposed to

manual gauging interventions. The paper presents an integrated methodology for

compensating errors detected using on-machine probing.In a multi-cut process,

intermittent probing which is achieved through replacing the cutting tool with a

touch probe, after each cut, can detect machining errors caused by deflection and

the tool offset error. A cutting compliance coefficient model is used to estimate

corrections to the tool path at the finishing cut based on a finite number of

measured errors at discrete locations for previous cuts. The model also anticipates

compliance changes and the effect of the compensated depth of cut. The complex

surface to be machined is represented by a B-Spline model. The compensated

tool path is obtained from B-Spline deformation techniques applied to the initial

tool path according to the discrete corrections. Milling tests are carried out with

and without compensation demonstrating a machining error reduction from +140

µm to ±20 µm.

Keywords: Milling; accuracy; on-machine measurement; error; compensation; B-

Spline deformation

1. Introduction

The conformity of a mechanical part to its drawing tolerance is an essential

requirement for a manufacturing process. The machined part accuracy is affected by

many error sources, some related to the machine-tool, others to the machining process.

Machine-tool systematic errors affect the positioning accuracy of the tool with respect

to the part. These are measured using a variety of instruments to detect the relative


displacement error of the tool with respect to the part while the machine is unloaded.

However, the systematic deviations related to the machining process, such as deflection

under the cutting forces and tool wear(Bandy 2001) cannot be investigated with this

strategy. It is believed that such errors can be measured directly by on machine

measurement (OMM). Using OMM, the machined part geometry can be measured

immediately after a cut without removing the part from its machining setup. One of the

principal advantages of OMM is the possible use of the measurement data to derive

corrective actions in order to improve the machining process accuracy for the current

part. As a result, a new task is added to the traditional mission of metrology so that it is

not used only to get the final part dimension but also to produce timely information

about the cutting process (Kunzmann et al. 2005). The process intermittent inspection

(PII) (Bandy 2001) provides inspection data for the part between cuts so that machining

errors detected at the semi-finishing cuts can be used to anticipate the errors at the

finishing cut and conduct tool path correction. The compensation model, developed

by(Guiassa and Mayer 2011), estimate the correction magnitude of the cutting tool

position for the finishing cut. At the semi-finishing cuts, the error and the cutting depth

are measured. The intermittent measurement, in multi-cut mode, produces data able to

estimate the trend of the cutting compliance coefficient which is used to predict the final

cut deviation. This prediction takes into a count the compensation and the material

removal effects on the machining error magnitude.

For free form surfaces, the discrete compensation vectors, computed at a limited

number of measured points locations according to the cutting compliance model, for a

multi-cut process (MCP)(Guiassa and Mayer 2011), must be used to associate a

deformation to the quasi-continuous free form tool path.


Free form deformation (FFD)(Sederberg and Parry 1986) remains a complex

task in engineering processes (Sarraga 2004, Biermann et al. 2010). In order to

compensate the anticipated geometric manufacturing errors, the initial geometry is

adjusted according to deformation vectors(Biermann et al. 2010). Such techniquesare

used, for example, tocompensate the anticipated spring-back defect in sheet metal

forming. The stamping tool geometry is modified so that the deviation is

compensated(Sarraga 2004). In machining, tool path modification was used to avoid

machine configurations inducingnumerical instability and uncontrollable positioning

due to inverse kinematic singularity problems (Affouard et al. 2004). The common

mathematical tools are the B-Spline and Bezier models for curves and surfaces. Both

Bezier and B-Spline are derived from NURBS and are used for their good mathematical

characteristics(Piegl and Tiller 1997, Barari et al. 2009). Typically, the FFD process

uses the geometry to be deformed and a set of deformation vectors to create the

deformed shape. The B-spine volume deformation process is a popular tool to deform a

given geometry according to deformation vectors (Biermann et al. 2010). In order to use

this process, the initial geometry is represented by a tensor product combination of three

B-Spline basis functions. For the B-Spline parameters (u,v,w) corresponds a point of

Cartesian coordinates (x, y, z) in the volume. The B-Spline parameters of the

deformation vector locations must be numerically computed for this model. The

deformation process is based on a least-squares minimization problem which finds the

control points adjustment. The objective is to reduce the deviation of the compensated

position from the model. This process requires a large number of deformation vectors

otherwise it can result in significant deviations(Sarraga 2004) which could compromise


the part accuracy. The proposed method does not require a minimum number of

measured points to fit the compensated geometry.

This paper is dedicated to adapt in milling process and apply the B-Spline

deformation techniques to compensate tool path, according to a restricted number of

compensation vectors. The compensation vectors are computed, at the measured point

locations, according to the cutting compliance model. Then, the data points used in the

fitting process are the uncompensated data adjusted by the compensation vectors.

2. Modeling and compensation of error detected using on-machine

measurement in a multi-cut process

2.1. Error model

The on-machine measured error is evaluated by comparing the actual surface

measurement data with the desired surface derived from the original tool path

commands. By using the machine tool as a measuring machine, thus substituting a touch

probe for the cutting tool, the machine tool detects the combined error due to tool offset

error (TOE) and machine-tool-part deflection (Bandy 2001). In a multi-cut process, as

shown in Figure 1, the final desired shapeshould be obtained after all cutting passes.

The error model (Guiassa and Mayer 2011), for the j-th position of the i-th cut, can be

written as:


Figure 1. Multi-cut process

jritoji ,, εεε += (1)

where toε is the tool offset error and

jijijri d ,,, ρε = (2)

is the error due to the effective compliance of the system. It is the deflection at the

position j under the cut i. The parameters ji ,ρ and jid , are the corresponding cutting

compliance coefficient (C3) and the actual depth of cut respectively. Along the cutting

profile, rε depends on the instantaneous local compliance of the system machine-tool-

part in the direction normal to the surface.

In order to compensate the error at the finishing cut using eqs 1 and 2, the TOE

and the actual C3 must be known at the finishing cut. In the two following sections, we

discuss the techniques proposed to measure the TOE and to predict the C3 for the

finishing cut.


2.2. On machine probe based technique for tool offset error estimation

Tool wear is one of the most important elements controlling the accuracy of the

tool geometry which affects the accuracy of the part. The TOE depends on the tool wear

evolution during the cutting process. Without compensation, the tool life, and so

production costs, depends on the tool wear limits(Astakhov 2004)required to produce

the part within the given tolerance.

Figure 2. Tool offset error estimation. According to the error model, it is the deviation

of the actual profile after the cut from the corresponding desired profile when the actual

depth of cut tends toward zero.


Figure 3. Linear estimation of TOE.

In this paper, a single value of TOE producing a constant offset, for the

complete machined profile, is considered. According to eqs.1 and 2, the cutting depth

error due to compliance is null when the actual depth of cut is zero and at this point, in

theory, the total error is equal to TOE. This is used to estimate TOE. Figure 2illustrates

the cutting path used to implement the TOE estimation procedure. By programming a

change in the depth of cut, the corresponding measured error is used to formulate the

linear approximation of toε .In order to respect the chip formation concept and the

necessary minimum chip thickness (Chae et al. 2006), the estimation of TOE can be

performed using the approximation illustrated in Figure 3 such that the minimum depth

of cut is viable.

2.3. Cutting compliance coefficient

The cutting compliance affects the accuracy generally by leaving extra material

on the part and depends on the machine, the tool, the part and the process. Theoretical

prediction of the system deflection is complicated because it is affected by the cutting

force, vibrations, the part/tool material and geometry. The proposed approach relies

solely on machine measurement through probing. The relation between the deflection


and the actual depth of cut, eq. 2, is used to predict the C3 directly from previous

measured errors and cutting depths.

Figure 4. Illustration of the C3 evolution, (a) general model, (b) linearized model.

In a multi-cut process, the deflection may increase from one cut to the next, even

for equal depths of cut, because the part loses stiffness after material removal.

Therefore, the C3 is affected by the material removal and increases as illustrated in

Figure 4-a. The parameters of ji ,ρ , jid , and TOE are measured through the intermittent

inspection process.

In order to compensate the system deflection at the finishing cut using the C3

method, the expected cutting compliance coefficient jf ,ρ is predicted before the final

cut and then is used to estimate the corresponding deflection. The Lagrange

extrapolation, illustrated in Figure 4-a, is adopted and is written as:

∑−

==

1

1,,,

f

ijijijf Lρρ (3)


where jiL , are the Lagrange basis polynomials associated with the cumulated depths,

after each cut(Guiassa and Mayer 2011) in the MCP, as:

∏∑∑

∑∑−

≠=

==

==

−

−=

1

,1

1,

1,

1,

1,

,

f

ikkk

sjs

i

sjs

k

sjs

f

sjs

ji

dd

dd

L (4)

Eqs. 1 and 2, at the finishing cut, can be written as:

jfjftojf d ,,, ρεε += (5)

where jfd , is the desired final cutting depth which includes the nominal (initial) depth

jnfd , and the previous machining error jf ,1−ε :

jfjnfjf dd ,1,, −+= ε (6)

In order to anticipate and eliminate the expected error, jf ,ε , this error is added to the tool

path in the opposite direction so that the compensation magnitude cj is:

jfjc ,ε= (7)

Note that, the coupled effect of compensation and deflection is taken into

account in the prediction of jf ,ε through the predicted C3 and the desired depth of cut

from eq.6.

As indicated previously, the radial compensation estimation is based on the C3

estimation for the final cut f. To reduce the number of inspected semi-finishing cut and


so doing increase productivity, the C3 estimation is linearized around the last semi-

finishing cut (f-1), as illustrated in Figure 4-b, resulting in the simplified relation:

jfjf

jfjfjfjf d

d,

,1

,2,1,1,

−

−−−

−+=

ρρρρ (8)

The compensation magnitude, for the measured position j, can be approximated for a

three cut process (3CP) with two inspected cut as(Guiassa and Mayer 2011):

( ) ( )12

2

2

32

2

3 εεεεε −

+−⋅+=

d

d

d

dc toto (9)

where 233 ε+= ndd is the desired cutting depth (eq. 6), 2d is the actual cutting depth for

the second cut and 2ε and 1ε are the measured errors after the second and the first cut

respectively.

3. Continuous tool path compensation from a set of discrete

compensation vectors

The goalof the continuous tool path compensation procedure is to generate the

compensated tool path from one initially programmed and a set of discrete

compensation vectors. Using the B-Spline deformation method(Biermann et al. 2010),

the compensated tool path is generated through a least-squares optimization problem to

create a B-Spline geometry close to the adjusted data points. In order to generate a

given B-Spline from data points, the number of fitted points is selected according to the

required fit quality(Piegl and Tiller 1997). Using OMM, the discrete compensation

vectors, needed to generate the adjusted data points, are calculated only where probing

data was gathered. These are kept to a minimum in order to reduce the loss of


production time due to the lengthy discrete probing process unless a faster scanning

process can be used.

The tool path compensation through the B-Spline deformation is performed

according to the following steps:

(1) the tool path is converted to a set of successive points and fitted to create a B-

Spline. The initial geometry must be retraced within the required tolerance;

(2) the set of points in (1) are adjusted according to the compensation vectors

computed at the measured points locations by eq.9;

(3) a new set of points is fitted to create the deformed B-Spline.

3.1.The tool path as a B-Spline

According to (Piegl and Tiller 1997), the B-Spline curve, )(uF , of degree n, is a

piecewise polynomial curve created such that:

]1,0[)()(0

, ∈=∑=

uuNun

iipi PF (10)

whereNi,pis the basis functions of degree p defined over a knot sequence

),...,,( 110 ++= pntttT . iP are the (n+1) control points. To approximate the data points

sQ ( tool path data points) by the B-Spline )(uF , the positions of the control points are

computed through the resolution of the minimization problem in the least-squares sense,

i.e.:

∑=

−=m

sss uJ

1

2)()(mini FQP

P (11)


where mszyx ssss ,...,1],[ ==Q , are the ordered data points coordinates to be

fitted. su is the pre-computed B-Spline parameter values corresponding to the data

points. The chord length parameterization is used (Piegl and Tiller 1997)to assign a

value su to each data points. Note that the initial G-code containing compact machining

cycles such as linear and circular interpolations must be rewritten to generate only small

linear interpolations so that the adjustment of the extracted points is possible.

3.2.Tool path adjustment strategy for B-Spline deformation

The initial data points are adjusted by adding small compensations sc ,

illustrated in Figure 5. For the point of B-Spline parameter value su , the adjusted point

sQ′ is computed as:

sss u cFQ +=′ )( (12)

where )( suF is the initial vector position given by the initial B-Spline function and

ssc n cs = is the adjustment vector with a magnitude of sc in the direction of the B-

Spline normal vector sn . sc is estimated as a linear combination of the two

compensations kc and 1+kc as:


Figure 5. Data points adjustment.

kk

kskkks

uu

uucccc

−−

−+=+

+1

1 )( (13)

where kc and 1+kc are the compensations computed at the two successive measured

points of parameter values ku and 1+ku such as 1+<< ksk uuu .

Note that the parameter value ju , for the measured point jM , is computed such

as: )(1jju MF −= , i.e. by projecting the measured points into the B-Spline(Piegl and

Tiller 1997).

The new set of points (sQ′ ) remain close to the initial tool path ( )( suF ). They are fitted

to create the deformed B-Spline, )(uF ′ as:

∑=

′=′n

iijpij uNu

0, )()( PF (14)

)(uF ′ has the same number of control points as )(uF and the same sequence of knot

vector. Therefore, only the positions of the control points are adjusted.


Figure 6. Flowchart of the NC program compensation.

The control points adjustment iP∆ , such as iii PPP ∆+=′ , are computed by

resolving a minimization problem similar to eq.11:


∑ ∑∑= ==∆

∆+⋅−′=′−′=∆m

si

n

iispis

m

sss uNuJ

1

2

0,

1

2)()()()(mini PPQFQP

P (15)

The modified B-Spline, F'(u), is the compensated tool path. The successive

compensated tool positions msuzuxuxu ssss ,...,1)],()()([)( =′′′=′F can be

generated by subdividing the compensated B-Spline into a set of linear segments with

respect to the chordal error tolerance to create the G-code with basic commands (G01).

The flowchart of the NC program compensation procedure is shown inFigure 6.

The tool path (successive tool positions) is extracted from the NC program as a

sequence of m ordered data points. The created B-Spline is close to the data points

within the given tolerance. The discrete compensations vectors jc are computed

according to the cutting compliance model for a multi-cut process and added to the tool

path at the positions of the B-Spline parameters ju . The compensated tool path is

generated by fitting the adjusted tool positions sQ′ .

4. Experimental validation

A machine-tool made by Huron, model KX8-Five fitted with a machine probe

model MP700 made by Renishaw is used to perform the milling and probing for the

tests. After roughing, the three-cut process and compensation procedure is conducted on

the part shown in Figure 7. Two identical parts are designed with a B-Spline curve and

machined, one without compensation and the other with compensation.


Figure 7. Test part.

All cutting parameters such as cutting speed and feed rates are kept constant for

both machining operations. The cutting tool is from HANITA model FZ 4AN3 19007.

Furthermore, in order to ensure good machine tool repeatability, both test pieces are

machined at the same machine tool workspace location.

Figure 8. Left, the locations of measured points along the 2D profile. Right, B-Spline

parameter values of the corresponding measured points.

After each cut, the probe replaces the cutting tool to perform the intermittent

measurement. The location of the probed points and the corresponding B-Spline

normalized parameters are shown in Figure 8. The parameter values are used to

program the probing operations.


Figure 9. On-machine measured error for three cut performed without compensation for

the test part.

Figure 10. Tool path planning for depth of cut reduction. At the zero programed depth,

the measured error is assumed equal to the TOE.


Figure 11. The discrete and the continuous (B-Spline deformation) compensation of the

tool path for the finishing cut of the 3CP performed for the second feature of the part in

Figure 7.

The on-machine measured error for the first feature is shown in Figure 9. The

error is an undercut material for all the profile and the magnitude is minimal at points

number 8 and 9. At these points, according to the error model eqs.1-2, the programmed

depth is reduced to minimize the compliance effect.This is used to estimate the TOE at

the zero programmed depth of cut. For this test parts, a fast and practical tool path

planning for TOE estimation is shown in Figure 10. The goal is to create a cutting zone

with a minimum cutting depth. The TOE is supposed equal to the measured error for

this zone. Following the tool path illustrated in Figure 10, a zero programed depth zone

is created. The TOE is approximated by the errors measured in this zone.

The tool offset measurement procedure is applied to each cut. The results show,

for this case study, that no significant change in the TOE occurs between cuts. This

justifies the use of a single value of TOE in the compensation formulation (eq. 10). As


can be seen in Figure 9, the evolution of the error in MCP is in agreement with the

thickness of the feature and the error increases significantly when the compliance

increases due to low thickness.

Figure 12. On-machine measured error for the 3CP with compensation for the finishing

cut performed for the second feature of the part shown in Figure 7.

For the second test piece, the results are shown in Figure 12. The OMM errors

for the first and the second cuts (uncompensated cuts) are similar to those obtained for

the first feature. The errors are reproduced within ±6 µm (95 %). For the finishing cut

(compensated cut), the tool path is corrected with the continuous compensation

computed through eq. 14 and shown in Figure 11. As shown in in Figure 12, the OMM

error is reduced from the maximum expected value of 140 µm (first featured) to less

than ±20 µm.

5. Conclusion

A model and its implementation strategy are proposed to compensate the

machining errors detected using an on-machine measurement procedure. In a multi-cut


milling process, the intermittent measurement data acquired at the semi-finishing cuts is

used to estimate the expected error at the finishing cut for the current part. This error is

attributed to the compliance effect and the tool offset error. The cutting compliance

model takes into account the previous measured error and anticipates the material

removal effect on the compliance. The tool path of the uncompensated finishing cut is

converted to a B-Spline function. The corresponding full compensated tool path is a free

form trajectory which is the uncompensated B-Spline deformed according to the

discrete compensations at a given set of measured points. The experiments show an

improvement of the machining accuracy. The error, as measured by the touch probe

fitted machine, is reduced, for the case study, from 140 µm in the uncompensated case

to ±20 µm after compensation.

Acknowledgments:

Authorswould like to thank CRIAQ and its industrial partners, PW corp. and Meloche Group,

for the financial support

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closed door machining-error compensation of complex surfaces

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